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# Probability

IB Math SL Topic 6

by

Tweet## Olivier Caron

on 22 April 2010#### Transcript of Probability

Topic 6: Statistics & Probability Probability distributions Binomial Distributions A type of discrete random variable which satisfies

4 conditions:

1)There are n identical trials

2)For each trial, there are 2 outcomes

3)The probability, p, remains constant throughout n trials

4)Trials are independent of each other

If the random variable X follows these criteria (with n trials and p probability outcome), the notation is

X~B(n,p) P(X=x) = (n nCr x ) p^x (1-p)^(n-x) where x=0,1,…….n Expected value Where X~B(n,p), E(X)=np Normal distribution Experimental Probabiliy trial: each repetition of the experiment

outcomes: different results possible ofr one trial of the experiment

frequency: number of timese an outcome occurs

relative frequency: frequency expressed as a precentage Sample Space Listing 2D Grid Tree diagram If the experiment is throwing a die

the sample space is {1,2,3,4,5,6} Theoretical Probability Defined as the measure of the chance that an event will occur Complimentary Events Probability Rules

P(A∩B) is the intersection AND the event

P(A∪B) is the union OR event

A’=1-A is the complement event (the NOT happening)

Laws of probability ddition law P(A∪B)=P(A)+P(B)-P(A∩B) Venn Diagram Mutually Exclusive Events events that have no common outcomes; where P(A∩B) = 0 P(A∪B)=P(A) + P(B) because P(A∩B) = 0 Conditionned Probability P(A|B)=[P(A∩B)]/P(B)

Where A will only happen given that B has already happened Independents VS Dependents Events Independent events are when the two events do not affect each other’s chances of occurring; this they are in different sample spaces.

Dependent events are when two events share a probability of occurring together and/or separate from each other. P(A|B)=P(A) or P(B|A)=P(B) P(A∩B)=P(A)P(B), where

P(A|B)=P(A)

= P(A∩B)/P(B)

= P(A)P(B)/P(B)

=P(A)

Proof X~N(μ,σ^2),

where μ is the mean and;

σ^2 is the standard deviation

Standardized value z is a transformation of the original data x to X. The formula for this is Where z=(x-μ)/σ Statistics-

The study of data and trend

Discrete Data are the facts that must be considered in units (eg. # of books, # of people)

Continuous Data facts that can be measured in fractions of a unit (eg. distance, time)

μ=(Σdata)/(# of data) This is the mean calculation

(average) Mode – the most frequent data value

Median – the middle data value in numerical order The nth value that is the midterm can be calculated with the equation

Midterm number = (n+1)/2

The interquartile range is the difference between the maximum and minimum values

Q3 – Q1 = IQR

NEED TO KNOW PERCENTILES!

Q1=25%

Q2=50%=mean

Q3=75%

The Box-and-whisker plot, also known as the

box plot is a graph that defines the minimum, Q1,

median, Q3, and maximum on a number line. Variance is the average of the squared deviation of each data value from the mean.

Standard Deviation is the square root of the variance. It is also the distance to which the mean deviates to represent a population.

68%

95% 98% Significance of standard deviation +/- 1 standard deviation : 68%

+/- 2 standard deviation : 95%

+/- 3 standard deviation : 98%

Full transcript4 conditions:

1)There are n identical trials

2)For each trial, there are 2 outcomes

3)The probability, p, remains constant throughout n trials

4)Trials are independent of each other

If the random variable X follows these criteria (with n trials and p probability outcome), the notation is

X~B(n,p) P(X=x) = (n nCr x ) p^x (1-p)^(n-x) where x=0,1,…….n Expected value Where X~B(n,p), E(X)=np Normal distribution Experimental Probabiliy trial: each repetition of the experiment

outcomes: different results possible ofr one trial of the experiment

frequency: number of timese an outcome occurs

relative frequency: frequency expressed as a precentage Sample Space Listing 2D Grid Tree diagram If the experiment is throwing a die

the sample space is {1,2,3,4,5,6} Theoretical Probability Defined as the measure of the chance that an event will occur Complimentary Events Probability Rules

P(A∩B) is the intersection AND the event

P(A∪B) is the union OR event

A’=1-A is the complement event (the NOT happening)

Laws of probability ddition law P(A∪B)=P(A)+P(B)-P(A∩B) Venn Diagram Mutually Exclusive Events events that have no common outcomes; where P(A∩B) = 0 P(A∪B)=P(A) + P(B) because P(A∩B) = 0 Conditionned Probability P(A|B)=[P(A∩B)]/P(B)

Where A will only happen given that B has already happened Independents VS Dependents Events Independent events are when the two events do not affect each other’s chances of occurring; this they are in different sample spaces.

Dependent events are when two events share a probability of occurring together and/or separate from each other. P(A|B)=P(A) or P(B|A)=P(B) P(A∩B)=P(A)P(B), where

P(A|B)=P(A)

= P(A∩B)/P(B)

= P(A)P(B)/P(B)

=P(A)

Proof X~N(μ,σ^2),

where μ is the mean and;

σ^2 is the standard deviation

Standardized value z is a transformation of the original data x to X. The formula for this is Where z=(x-μ)/σ Statistics-

The study of data and trend

Discrete Data are the facts that must be considered in units (eg. # of books, # of people)

Continuous Data facts that can be measured in fractions of a unit (eg. distance, time)

μ=(Σdata)/(# of data) This is the mean calculation

(average) Mode – the most frequent data value

Median – the middle data value in numerical order The nth value that is the midterm can be calculated with the equation

Midterm number = (n+1)/2

The interquartile range is the difference between the maximum and minimum values

Q3 – Q1 = IQR

NEED TO KNOW PERCENTILES!

Q1=25%

Q2=50%=mean

Q3=75%

The Box-and-whisker plot, also known as the

box plot is a graph that defines the minimum, Q1,

median, Q3, and maximum on a number line. Variance is the average of the squared deviation of each data value from the mean.

Standard Deviation is the square root of the variance. It is also the distance to which the mean deviates to represent a population.

68%

95% 98% Significance of standard deviation +/- 1 standard deviation : 68%

+/- 2 standard deviation : 95%

+/- 3 standard deviation : 98%